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Non-contact technology for monitoring multiple people's vital signs, such as respiration and heartbeat, has been investigated in recent years due to the rising cardiopulmonary morbidity, the risk of transmitting diseases, and the heavy burden on the medical staff. Frequency modulated continuous wave (FMCW) radars have shown great promise in meeting these needs. However, contemporary techniques for non-contact vital signs monitoring (NCVSM) via FMCW radars, are based on simplistic models, and present difficulties coping with noisy environments containing multiple objects. In this work, we develop an extended model of FMCW radar signals in a noisy setting containing multiple people and clutter. By utilizing the sparse nature of the modeled signals in conjunction with human-typical cardiopulmonary features, we can accurately localize humans and reliably monitor their vital signs, using only a single channel and a single-input-single-output setup. To this end, we first show that spatial sparsity allows for both accurate detection of multiple people and computationally efficient extraction of their Doppler samples, using a joint sparse recovery approach. Given the extracted samples, we develop a method named Vital Signs based Dictionary Recovery (VSDR), which uses a dictionary-based approach to search for the desired rates of respiration and heartbeat over high-resolution grids corresponding to normal cardiopulmonary activity. The advantages of the proposed method are illustrated through examples that combine the proposed model with real data of $30$ monitored individuals. We demonstrate accurate human localization in a clutter-rich scenario that includes both static and vibrating objects, and show that our VSDR approach outperforms existing techniques, based on several statistical metrics. The findings support the widespread use of FMCW radars with the proposed algorithms in healthcare.